New plane-sweep algorithms for distance-based join queries in spatial databases

Efficient and effective processing of the distance-based join query (DJQ) is of great importance in spatial databases due to the wide area of applications that may address such queries (mapping, urban planning, transportation planning, resource management, etc.). The most representative and studied DJQs are the K Closest Pairs Query (KCPQ) and εDistance Join Query (εDJQ). These spatial queries involve two spatial data sets and a distance function to measure the degree of closeness, along with a given number of pairs in the final result (K) or a distance threshold (ε). In this paper, we propose four new plane-sweep-based algorithms for KCPQs and their extensions for εDJQs in the context of spatial databases, without the use of an index for any of the two disk-resident data sets (since, building and using indexes is not always in favor of processing performance). They employ a combination of plane-sweep algorithms and space partitioning techniques to join the data sets. Finally, we present results of an extensive experimental study, that compares the efficiency and effectiveness of the proposed algorithms for KCPQs and εDJQs. This performance study, conducted on medium and big spatial data sets (real and synthetic) validates that the proposed plane-sweep-based algorithms are very promising in terms of both efficient and effective measures, when neither inputs are indexed. Moreover, the best of the new algorithms is experimentally compared to the best algorithm that is based on the R-tree (a widely accepted access method), for KCPQs and εDJQs, using the same data sets. This comparison shows that the new algorithms outperform R-tree based algorithms, in most cases.     

Approximate Comparison of Functions Computed by Distance Automata

Distance automata are automata weighted over the semiring \((\mathbb {N}\cup \{\infty \},\min , +)\) (the tropical semiring). Such automata compute functions from words to \(\mathbb {N}\cup \{\infty \}\). It is known from Krob that the problems of deciding ‘ f≤g’ or ‘ f=g’ for f and g computed by distance automata is an undecidable problem. The main contribution of this paper is to show that an approximation of this problem is decidable. We present an algorithm which, given ε>0 and two functions f,g computed by distance automata, answers “yes” if f≤(1?ε)g, “no” if f≦?g, and may answer “yes” or “no” in all other cases. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation of the closure under products of sets of matrices over the tropical semiring. Lastly, our theorem of affine domination gives better bounds on the precision of known decision procedures for cost automata, when restricted to distance automata.     

Improving configuration checking for satisfiable random <Emphasis Type="Italic">k</Emphasis>-SAT instances

Local search algorithms based on the Configuration Checking (CC) strategy have been shown to be efficient in solving satisfiable random k-SAT instances. The purpose of the CC strategy is to avoid the cycling problem, which corresponds to revisiting already flipped variables too soon. It is done by considering the neighborhood of the formula variables. In this paper, we propose to improve the CC strategy on the basis of an empirical study of a powerful algorithm using this strategy. The first improvement introduces a new and simple criterion, which refines the selection of the variables to flip for the 3-SAT instances. The second improvement is achieved by using the powerful local search algorithm Novelty with the adaptive noise setting. This algorithm enhances the efficiency of the intensification and diversification phases when solving k-SAT instances with k ≥ 4. We name the resulting local search algorithm Ncca+ and show its effectiveness when treating satisfiable random k-SAT instances issued from the SAT Challenge 2012. Ncca+ won the bronze medal of the random SAT track of the SAT Competition 2013.     

On a new extension of BTP for binary CSPs

The study of broken-triangles is becoming increasingly ambitious, by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through either value merging or variable elimination. Considerable progress has been made in extending this important concept, such as dual broken-triangle and weakly broken-triangle, in order to maximize the number of captured tractable CSP instances and/or the number of merged values. Specifically, m-wBTP allows us to merge more values than BTP. DBTP, ??-BTP, k-BTP, WBTP and m-wBTP permit us to capture more tractable instances than BTP. However, except BTP, none of these extensions allows variable elimination while preserving satisfiability. Moreover, k-BTP and m-wBTP define bigger tractable classes around BTP but both of them generally need a high level of consistency. Here, we introduce a new weaker form of BTP, called m-fBTP for flexible broken-triangle property, which will represent a compromise between most of these previous tractable properties based on BTP. m-fBTP allows us on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define a new bigger tractable class for which arc consistency is a decision procedure. Likewise, m-fBTP permits to merge more values than BTP but fewer than m-wBTP. The binary CSP instances satisfying m-fBTP are solved by algorithms of the state-of-the-art like MAC and RFL in polynomial time. An open question is whether it is possible to compute, in polynomial time, the existence of some variable ordering for which a given instance satisfies m-fBTP.     

A commentary on “Hybrid search for minimal perturbation in Dynamic CSPs”

The Hybrid Search for Minimal Perturbation Problems algorithm in Dynamic CSP (HS_MPP) (Zivan, Constraints, 16(3), 228–249, 2011) ensures for a given dynamic problem and its solution to the previous CSP, to find the optimal solution to the newly generated CSP. This proposed method exploits the fact that its reported solution must satisfy two requirements. First, that it is a solution for T complete assignment for the derived CSP and second, that it is as close as possible to the solution of the former CSP. Unfortunately, the pseudo-code of the algorithm in Zivan (Constraints, 16(3), 228–249, 2011) is confusing and may lead to an implementation in which HS_MPP may not perform the expected outcomes of a given instance of Dynamic CSPs correctly. In this erratum, we demonstrate the possible undesired outcomes and give corrections in HS_MPP’s pseudo-code.     

Exact and approximate flexible aggregate similarity search

Aggregate similarity search, also known as aggregate nearest-neighbor (Ann) query, finds many useful applications in spatial and multimedia databases. Given a group Q of M query objects, it retrieves from a database the objects most similar to Q, where the similarity is an aggregation (e.g., \({{\mathrm{sum}}}\), \(\max \)) of the distances between each retrieved object p and all the objects in Q. In this paper, we propose an added flexibility to the query definition, where the similarity is an aggregation over the distances between p and any subset of \(\phi M\) objects in Q for some support \(0< \phi \le 1\). We call this new definition flexible aggregate similarity search and accordingly refer to a query as a flexible aggregate nearest-neighbor ( Fann ) query. We present algorithms for answering Fann queries exactly and approximately. Our approximation algorithms are especially appealing, which are simple, highly efficient, and work well in both low and high dimensions. They also return near-optimal answers with guaranteed constant-factor approximations in any dimensions. Extensive experiments on large real and synthetic datasets from 2 to 74 dimensions have demonstrated their superior efficiency and high quality.     

Making Doubling Metrics Geodesic

The starting point of our research is the following problem: given a doubling metric ?=(V,d), can one (efficiently) find an unweighted graph G′=(V′,E′) with V?V′ whose shortest-path metric d′ is still doubling, and which agrees with d on V×V? While it is simple to show that the answer to the above question is negative if distances must be preserved exactly. However, allowing a (1+ε) distortion between d and d′ enables us bypass this hurdle, and obtain an unweighted graph G′ with doubling dimension at most a factor O(log?ε ^?1) times the doubling dimension of G.More generally, this paper gives algorithms that construct graphs G′ whose convex (or geodesic) closure has doubling dimension close to that of ?, and the shortest-path distances in G′ closely approximate those of ? when restricted to V×V. Similar results are shown when the metric ? is an additive (tree) metric and the graph G′ is restricted to be a tree.     

Mathematical models and routing algorithms for CAD technological preparation of cutting processes

Resource-conscious technologies for cutting sheet material include the ICP and ECP technologies that allow for aligning fragments of the contours of cutouts. In this work, we show the mathematical model for the problem of cutting out parts with these technologies and algorithms for finding cutting tool routes that satisfy technological constraints. We give a solution for the problem of representing a cutting plan as a plane graph G = (V,F,E), which is a homeomorphic image of the cutting plan. This has let us formalize technological constraints on the trajectory of cutting the parts according to the cutting plan and propose a series of algorithms for constructing a route in the graph G = (V,F,E), which is an image of an admissible trajectory. Using known coordinates of the preimages of vertices of graph G = (V,F,E) and the locations of fragments of the cutting plan that are preimages of edges of graph G = (V,F,E), the resulting route in the graph G = (V,E) can be interpreted as the cutting tool’s trajectory.The proposed algorithms for finding routes in a connected graph G have polynomial computational complexity. To find the optimal route in an unconnected graph G, we need to solve, for every dividing face f of graph G, a travelling salesman problem on the set of faces incident to f.     

Indefinite summation of rational functions with additional minimization of the summable part

An algorithm of indefinite summation of rational functions is proposed. For a given function f(x), it constructs a pair of rational functions g(x) and r(x) such that f(x) = g(x + 1) ? g(x) + r(x), where the degree of the denominator of r(x) is minimal, and, when this condition is satisfied, the degree of the denominator of g(x) is also minimal.     

Testing Convexity Properties of Tree Colorings

A coloring of a graph is convex if it induces a partition of the vertices into connected subgraphs. Besides being an interesting property from a theoretical point of view, tests for convexity have applications in various areas involving large graphs. We study the important subcase of testing for convexity in trees. This problem is linked, among other possible applications, with the study of phylogenetic trees, which are central in genetic research, and are used in linguistics and other areas. We give a 1-sided, non-adaptive, distribution-free ε-test for the convexity of tree colorings. The query complexity of our test is O(k/ε), where k is the number of colors, and the additional computational complexity is O(n). On the other hand, we prove a lower bound of \(\Omega(\sqrt{k/\epsilon})\) on the query complexity of tests for convexity in the standard model, which applies even for (unweighted) paths. We also consider whether the dependency on k can be reduced in some cases, and provide an alternative testing algorithm for the case of paths. Then we investigate a variant of convexity, namely quasi-convexity, in which all but one of the colors are required to induce connected components. For this problem we provide a 1-sided, non-adaptive ε-test with query complexity O(k/ε ²) and time complexity O(kn/ε). For both our convexity and quasi-convexity tests, we show that, assuming that a query takes constant time, the time complexity can be reduced to a constant independent of n if we allow a preprocessing stage of time O(n) and O(n ²), respectively. Finally, we show how to test for a variation of convexity and quasi-convexity where the maximum number of connectivity classes of each color is allowed to be a constant value other than 1.     

Translation-based approaches for solving disjunctive temporal problems with preferences

Disjunctive Temporal Problems (DTPs) with Preferences (DTPPs) extend DTPs with piece-wise constant preference functions associated to each constraint of the form l ≤ x ? y ≤ u, where x,y are (real or integer) variables, and l,u are numeric constants. The goal is to find an assignment to the variables of the problem that maximizes the sum of the preference values of satisfied DTP constraints, where such values are obtained by aggregating the preference functions of the satisfied constraints in it under a “max” semantic. The state-of-the-art approach in the field, implemented in the native DTPP solver Maxilitis, extends the approach of the native DTP solver Epilitis. In this paper we present alternative approaches that translate DTPPs to Maximum Satisfiability of a set of Boolean combination of constraints of the form l?x ? y?u, ? ∈{<,≤}, that extend previous work dealing with constant preference functions only. We prove correctness and completeness of the approaches. Results obtained with the Satisfiability Modulo Theories (SMT) solvers Yices and MathSAT on randomly generated DTPPs and DTPPs built from real-world benchmarks, show that one of our translation is competitive to, and can be faster than, Maxilitis (This is an extended and revised version of Bourguet et al. 2013).     

Property-Preserving Data Reconstruction

We initiate a new line of investigation into online property-preserving data reconstruction. Consider a dataset which is assumed to satisfy various (known) structural properties; e.g., it may consist of sorted numbers, or points on a manifold, or vectors in a polyhedral cone, or codewords from an error-correcting code. Because of noise and errors, however, an (unknown) fraction of the data is deemed unsound, i.e., in violation with the expected structural properties. Can one still query into the dataset in an online fashion and be provided data that is always sound? In other words, can one design a filter which, when given a query to any item I in the dataset, returns a sound item J that, although not necessarily in the dataset, differs from I as infrequently as possible. No preprocessing should be allowed and queries should be answered online.We consider the case of a monotone function. Specifically, the dataset encodes a function f:{1,…,n}?? R that is at (unknown) distance ε from monotone, meaning that f can—and must—be modified at ε n places to become monotone.Our main result is a randomized filter that can answer any query in O(log?² nlog? log?n) time while modifying the function f at only O(ε n) places. The amortized time over n function evaluations is O(log?n). The filter works as stated with probability arbitrarily close to 1. We provide an alternative filter with O(log?n) worst case query time and O(ε nlog?n) function modifications. For reconstructing d-dimensional monotone functions of the form f:{1,…,n} ^d ? ? R, we present a filter that takes (2 ^O(d)(log?n)^4d?2log?log?n) time per query and modifies at most O(ε n ^d ) function values (for constant d).     

<Emphasis Type="Italic">D</Emphasis>-decomposition technique state-of-the-art

It is a survey of recent extensions and new applications for the classical D-decomposition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The D-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given H _∞-criterion. It is shown that the D-decomposition technique can be naturally linked with M-Δ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via D-decomposition technique.     

Efficient <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript> norm based sparse subspace clustering via smooth IRLS and ADMM

Recently, sparse subspace clustering, as a subspace learning technique, has been successfully applied to several computer vision applications, e.g. face clustering and motion segmentation. The main idea of sparse subspace clustering is to learn an effective sparse representation that are used to construct an affinity matrix for spectral clustering. While most of existing sparse subspace clustering algorithms and its extensions seek the forms of convex relaxation, the use of non-convex and non-smooth l _q (0 < q < 1) norm has demonstrated better recovery performance. In this paper we propose an l _q norm based Sparse Subspace Clustering method (lqSSC), which is motivated by the recent work that l _q norm can enhance the sparsity and make better approximation to l ₀ than l ₁. However, the optimization of l _q norm with multiple constraints is much difficult. To solve this non-convex problem, we make use of the Alternating Direction Method of Multipliers (ADMM) for solving the l _q norm optimization, updating the variables in an alternating minimization way. ADMM splits the unconstrained optimization into multiple terms, such that the l _q norm term can be solved via Smooth Iterative Reweighted Least Square (SIRLS), which converges with guarantee. Different from traditional IRLS algorithms, the proposed algorithm is based on gradient descent with adaptive weight, making it well suit for general sparse subspace clustering problem. Experiments on computer vision tasks (synthetic data, face clustering and motion segmentation) demonstrate that the proposed approach achieves considerable improvement of clustering accuracy than the convex based subspace clustering methods.     

On Detection of Gaussian Stochastic Sequences

We consider the minimax detection problem for a Gaussian random signal vector in white Gaussian additive noise. It is assumed that an unknown vector σ of signal vector intensities belongs to a given set ε. We investigate when it is possible to replace the set ε with a smaller set ε₀ without loss of quality (and, in particular, replace it with a single point σ₀).     

Approximating the Metric TSP in Linear Time

Given a metric graph G=(V,E) of n vertices, i.e., a complete graph with a non-negative real edge cost function satisfying the triangle inequality, the metricity degree of G is defined as \(\beta=\max_{x,y,z\in V}\{\frac{c(x,y)}{c(x,z)+c(y,z)}\}\in[\frac{1}{2},1]\). This value is instrumental to establish the approximability of several NP-hard optimization problems definable on G, like for instance the prominent traveling salesman problem, which asks for finding a Hamiltonian cycle of G of minimum total cost. In fact, this problem can be approximated quite accurately depending on the metricity degree of G, namely by a ratio of either \(\frac{2-\beta}{3(1-\beta)}\) or \(\frac{3\beta^{2}}{3\beta^{2}-2\beta+1}\), for \(\beta<\frac{2}{3}\) or \(\beta\geq \frac{2}{3}\), respectively. Nevertheless, these approximation algorithms have O(n ³) and O(n ^2.5log?^1.5 n) running time, respectively, and therefore they are superlinear in the Θ(n ²) input size. Thus, since many real-world problems are modeled by graphs of huge size, their use might turn out to be unfeasible in practice, and alternative approaches requiring only O(n ²) time are sought. However, with this restriction, all the currently available approaches can only guarantee a 2-approximation ratio for the case β=1, which means a \(\frac{2\beta^{2}}{2\beta^{2}-2\beta+1}\)-approximation ratio for general β<1. In this paper, we show how to elaborate—without affecting the space and time complexity—one of these approaches, namely the classic double-MST heuristic, in order to obtain a 2β-approximate solution. This improvement is effective, since we show that the double-MST heuristic has in general a performance ratio strictly larger than 2β, and we further show that any alternative elaboration of it cannot lead to a performance ratio better than 2β?ε, for any ε>0. Our theoretical results are complemented with an extensive series of experiments, that show the practical appeal of our approach.     

Cache-Oblivious R-Trees

We develop a cache-oblivious data structure for storing a set S of N axis-aligned rectangles in the plane, such that all rectangles in S intersecting a query rectangle or point can be found efficiently. Our structure is an axis-aligned bounding-box hierarchy and as such it is the first cache-oblivious R-tree with provable performance guarantees. If no point in the plane is contained in more than a constant number of rectangles in S, we can construct, for any constant ε, a structure that answers a rectangle query using \(O(\sqrt{N/B}+T/B)\) memory transfers and a point query using O((N/B) ^ε ) memory transfers, where T is the number of reported rectangles and B is the block size of memory transfers between any two levels of a multilevel memory hierarchy. We also develop a variant of our structure that achieves the same performance on input sets with arbitrary overlap among the rectangles. The rectangle query bound matches the bound of the best known linear-space cache-aware structure.     

Bounded Unpopularity Matchings

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M′ such that more people prefer M′ to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by Abraham et al. (SIAM J. Comput. 37(4):1030–1045, 2007). If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity—unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching M with the minimum value of u(M) or g(M) is NP-hard, and that if G does not admit a popular matching, then we have u(M)≥2 for all matchings M in G.Here we show that a matching M that achieves u(M)=2 can be computed in \(O(m\sqrt{n})\) time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H=H ₂,H ₃,…,H _k such that if H _k admits a matching that matches all people, then we can compute in \(O(km\sqrt{n})\) time a matching M such that u(M)≤k?1 and \(g(M)\le n(1-\frac{2}{k})\). Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.     

On Optimally Partitioning a Text to Improve Its Compression

In this paper we investigate the problem of partitioning an input string T in such a way that compressing individually its parts via a base-compressor C gets a compressed output that is shorter than applying C over the entire T at once. This problem was introduced in Buchsbaum et al. (Proc. of 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 175–184, 2000; J. ACM 50(6):825–851, 2003) in the context of table compression, and then further elaborated and extended to strings and trees by Ferragina et al. (J. ACM 52:688–713, 2005; Proc. of 46th IEEE Symposium on Foundations of Computer Science, pp. 184–193, 2005) and Mäkinen and Navarro (Proc. of 14th Symposium on String Processing and Information Retrieval, pp. 229–241, 2007). Unfortunately, the literature offers poor solutions: namely, we know either a cubic-time algorithm for computing the optimal partition based on dynamic programming (Buchsbaum et al. in J. ACM 50(6):825–851, 2003; Giancarlo and Sciortino in Proc. of 14th Symposium on Combinatorial Pattern Matching, pp. 129–143, 2003), or few heuristics that do not guarantee any bounds on the efficacy of their computed partition (Buchsbaum et al. in Proc. of 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 175–184, 2000; J. ACM 50(6):825–851, 2003), or algorithms that are efficient but work in some specific scenarios (such as the Burrows-Wheeler Transform, see e.g. Ferragina et al. in J. ACM 52:688–713, 2005; Mäkinen and Navarro in Proc. of 14th Symposium on String Processing and Information Retrieval, pp. 229–241, 2007) and achieve compression performance that might be worse than the optimal-partitioning by a Ω(log?n/log?log?n) factor. Therefore, computing efficiently the optimal solution is still open (Buchsbaum and Giancarlo in Encyclopedia of Algorithms, pp. 939–942, 2008). In this paper we provide the first algorithm which computes in O(nlog?_1+ε n) time and O(n) space, a partition of T whose compressed output is guaranteed to be no more than (1+ε)-worse the optimal one, where ε may be any positive constant fixed in advance. This result holds for any base-compressor C whose compression performance can be bounded in terms of the zero-th or the k-th order empirical entropy of the text T. We will also discuss extensions of our results to BWT-based compressors and to the compression booster of Ferragina et al. (J. ACM 52:688–713, 2005).     

Finding Coefficients of the Full Array of Motion-Independent <Emphasis Type="Italic">N</Emphasis>-Body Potentials of Metric Gravity from Gravity’s Exterior and Interior Effacement Algebra

In the author’s previous publications, a recursive linear algebraic method was introduced for obtaining (without gravitational radiation) the full potential expansions for the gravitational metric field components and the Lagrangian for a general N-body system. Two apparent properties of gravity— Exterior Effacement and Interior Effacement—were defined and fully enforced to obtain the recursive algebra, especially for the motion-independent potential expansions of the general N-body situation. The linear algebraic equations of this method determine the potential coefficients at any order n of the expansions in terms of the lower-order coefficients. Then, enforcing Exterior and Interior Effacement on a selecedt few potential series of the full motion-independent potential expansions, the complete exterior metric field for a single, spherically-symmetric mass source was obtained, producing the Schwarzschild metric field of general relativity. In this fourth paper of this series, the complete spatial metric’s motion-independent potentials for N bodies are obtained using enforcement of Interior Effacement and knowledge of the Schwarzschild potentials. From the full spatial metric, the complete set of temporal metric potentials and Lagrangian potentials in the motion-independent case can then be found by transfer equations among the coefficients κ(n, α) → λ(n, ε) → ξ(n, α) with κ(n, α), λ(n, ε), ξ(n, α) being the numerical coefficients in the spatial metric, the Lagrangian, and the temporal metric potential expansions, respectively.